Find the Inverse x^(1/2)

$x^{\frac{1}{2}}$

Step 1

Interchange the variables.

$x = y^{\frac{1}{2}}$

Step 2

Solve for $y$ .

Rewrite the equation as $y^{\frac{1}{2}} = x$ .

$y^{\frac{1}{2}} = x$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

$y^{1} = x^{2}$

Simplify.

$y = x^{2}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = x^{2}$

Step 4

Verify if $f^{- 1} (x) = x^{2}$ is the inverse of $f (x) = x^{\frac{1}{2}}$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (x^{\frac{1}{2}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (x^{\frac{1}{2}}) = {(x^{\frac{1}{2}})}^{2}$

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (x^{\frac{1}{2}}) = x^{\frac{1}{2} \cdot 2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (x^{\frac{1}{2}}) = x^{\frac{1}{2} \cdot 2}$

Rewrite the expression.

$f^{- 1} (x^{\frac{1}{2}}) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (x^{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{2}) = {(x^{2})}^{\frac{1}{2}}$

Multiply the exponents in ${(x^{2})}^{\frac{1}{2}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (x^{2}) = x^{2 (\frac{1}{2})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (x^{2}) = x^{2 (\frac{1}{2})}$

Rewrite the expression.

$f (x^{2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{2}$ is the inverse of $f (x) = x^{\frac{1}{2}}$ .

$f^{- 1} (x) = x^{2}$

Find the Inverse x^(1/2)

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